This test case was suggested by Williamson et al. [47] (Problem #1). A "cosine bell" initial concentration is given by
h(λ,θ)=(h_{0}/2)(1+cos(π r/R)) |
if r<R and 0 otherwise, with R=1/3 and
r=arccos[sinθ_{c}sinθ+cosθ_{c}cosθcos(λ−λ_{c})] |
the great circle distance between longitude, latitude (λ,θ) and the center initially taken as (λ_{c},θ_{c})=(3π/2,0).
The advection velocity field corresponds to solid-body rotation at an angle α to the polar axis of the spherical coordinate system. It is given by the streamfunction
ψ=−u_{0}(sinθcosα−cosλcosθsinα) |
The cosine bell field is rotated once around the sphere and should come back exactly to its original position. The difference between the initial and final fields is a measure of the accuracy of the advection scheme coupled with the spherical coordinate mapping (the "conformal expanded spherical cube" metric in our case).
For the "spherical cube" metric, two angles are considered: 45 degrees which rotates the cosine bell above four of the eight "poles" of the mapping and 90 degrees which avoids the poles entirely. Mass is conserved to within machine accuracy in either case.
The mesh is adapted dynamically according to the gradient of tracer concentration.
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